Kerr–Schild Perturbations
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Kerr–Schild perturbations are a special type of
perturbation Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbatio ...
to a
spacetime metric In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The me ...
which only appear linearly in the Einstein field equations which describe
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
. They were found by
Roy Kerr Roy Patrick Kerr (; born 16 May 1934) is a New Zealand mathematician who discovered the Kerr geometry, an exact solution to the Einstein field equation of general relativity. His solution models the gravitational field outside an uncharged ...
and
Alfred Schild Alfred Schild (September 7, 1921 – May 24, 1977) was a leading Austrian American physicist, well known for his contributions to the Golden age of general relativity (1960–1975). Biography Schild was born in Istanbul on September 7, 1921. His ...
in 1965.


Form

A generalised Kerr–Schild perturbation has the form h_=V l_a l_b, where V is a scalar and l_a is a null vector with respect to the background spacetime. It can be shown that any perturbation of this form will only appear quadratically in the Einstein equations, and only linearly if the condition l^a \nabla_a l_b =\phi l_b, where \phi is a scalar, is imposed. This condition is equivalent to requiring that the orbits of l^a are geodesics.


Applications

While the form of the perturbation may appear very restrictive, there are several black hole metrics which can be written in Kerr–Schild form, such as
Schwarzschild Schwarzschild () is a German surnameIt is likely to be misspelled and/or mispronounced by native English speakers, particularly involving failure to grasp that * German ''sch'' (at the beginning of ''each'' of the two syllables) is pronounced as E ...
(stationary black hole),
Kerr Kerr may refer to: People * Kerr (surname) * Kerr (given name) Places ;United States * Kerr Township, Champaign County, Illinois * Kerr, Montana, A US census-designated place * Kerr, Ohio, an unincorporated community *Kerr County, Texas Other ...
(rotating), Reissner–Nordström (charged) and Kerr–Newman (both charged and rotating).


References

General relativity {{relativity-stub